I borrowed Liboff’s book from the public library again, figuring I’d take a second look. My initial impression of this book was that it was garbage, at least for a new learner.
This time around I know a bit more. I’ve now listened to Susskind’s QM lectures, which give a top down approach that turns out to be similar to Liboff’s (both pulling the approach out of thin air and magic hats instead of trying to show a logical progression).
The real key to understanding things was a good read of Bohm’s Quantum theory book, in particular chapters 3 and 9, and I’ve blogged about those separately.
Complementing Bohm’s book I’ve read the first two chapters of Pauli’s wave mechanics book.
Pauli’s book isn’t particularily easy to follow, but there are a number of aspects of it that I do like. One is the mini relativity primer in one page. He really knows that subject. This book is similar to Bohm’s in that all the integrals are written out in full. It has an old fashioned-ness that gets the core ideas across in a straightforward way.
I liked the way Pauli motivates the non-relativistic wave equation by making a approximation starting from the relativistic scalar wave equation. That approximation ends up with an extra term that the normal Schrodinger equation doesn’t have, and later when he switches to the regular non-relativistic Schrodinger equation he just drops this term. It will be interesting once I learn about the Dirac equation to see how the transition between the Klien Gordon and that equation works.
Pauli’s treatment of the uncertaintly relation is direct and clever and a classic example of working backwards from an answer to get to the solution in the fastest possible way. The neat thing about his way of doing it is how easily it shows how equality follows only for the Gaussian distribution.
I have to admit that I didn’t fully understand his measuring arrangement examples (the microscope and coherency parts in particular). Perhaps I got hung up on not understanding where the Abbe sine condition came from. The relativistic particle and light wave collision example is quite neat, but also highlights the fact that I really have to go and give that Hamilton’s equation treatment in Goldstein a good read. He used those ideas in his relativity review as well.
In Chapter II of Pauli’s book he has got a pretty slick treatment of 3D eigenfunction orthogonality for a boundary where the probability current is zero.
His completeness relation coverage was not particularily easy to follow. I wrote up my own notes on this separately, and managed to make sense of it for myself.
The final bits in chapter II, the “initial value problem and fundamental solution” I’m going to have to come back to. I can follow it step by step, but it’s not particularily easy. The end result makes sense. He comes up with a Gaussian form of the delta function by considering an impulse response. I’m not sure exactly how to relate all that to things that I currently know, and will probably have to try to work this problem myself at my own pace before I’ll really get it. Perhaps to make it different and avoid just spitting out exactly what he did I should try it myself in 3D.
I had intended to drop a few thoughts about Liboff’s book and instead dug out all my thoughts on Pauli’s chapter 1 and 2. Ah well. So, back to Liboff…
In this second look, after picking up some motivation and details in other sources, I rather like what I’ve read so far. It’s true that he has absolutely no attempt to motivate the ideas, but if you get that elsewhere what he does cover is done well. What did I like about his book?
The one dimensional momentum operator, so well motivated by Bohm, and so randomly stated by Liboff is actually a nice example of a one dimensional wave equation (ie: when stated in eigenvalue equation form). How he uses this as a simple example of the Helmholtz equation is also nice. That you can get the Helmholtz equation from separation of variables of the regular mechanical (or E&M) wave equation as well as from the QM wave equation makes some sense, so it’s nice to see this pointed out to see commonalities between the different parts physics.
Liboff’s treatment of the initial value problem is very nice. Much easier to understand than Pauli’s delta function variant and it shows a concrete example of a unitary operator that clarifies what Susskind was getting at in his lectures. Here the unitary operator (although it wasn’t named as such) is expressed as an integrating factor, and encodes all the time evolution of the initial state. Quite slick, and his example where he considers the eigenstates of the Hamiltonian really firms it up and takes the abstraction out of things.
Some notes on problems in Liboff:
Problem 3.5 is kind of nice, put a number to the energy. I calculated a value of about 150 eV for the 10^-8 cm wavelength electron. Wonder if I got that right? I got lazy with units and did some of the calculation with google calculator.
Problem 3.6 on the delta function properties. I’m unsure about how to show (b). . The others I did, but didn’t see how to do (e) until tacking the more general and abstract varient of it (i), which is amusing.
Problems 3.16-3.23 (all the end of chapter problems). Have only done the first so far (that one I could do in my head without paper while I was in the hot tub;) The rest will have to wait for tommorrow.